Lusin’s condition (N) and mappings with nonnegative Jacobians. (English) Zbl 0807.46032

A continuous mapping \(f: G\to\mathbb{R}^ n\) (\(G\) a domain in \(\mathbb{R}^ n)\) is said to satisfy Lusins’ condition (N) if \(f(A)\) has zero Lebesgue measure whenever \(A\subseteq G\) has zero Lebesgue measure. The author investigates Lusins’ condition for continuous functions in \(W^{1,n} (G,\mathbb{R}^ n)\), noting that in \(W^{1,p} (G,\mathbb{R}^ n)\) with \(p>n\) this condition holds while for \(p<n\) it generally fails. The results are in close connection with those of Yu. G. Reshetnyak. Among other results, the author proves that:
For \(f\in W^{1,n} (G,\mathbb{R}^ n)\) a continuous function with \(Jf\geq 0\) a.e., the condition (N) is equivalent with the Sard condition (i.e. \(Jf(x)=0\) a.e. on an open set \(A\subseteq G\), yields \(\text{meas } f(A) =0\)).
For a continuous \(f\in W^{1,2} (G,\mathbb{R}^ 2)\), the condition (N) is equivalent with \(f\) almost open in \(G\).


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26B10 Implicit function theorems, Jacobians, transformations with several variables
55M25 Degree, winding number
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